Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909456417161216 |
|---|---|
| author | Ma, Yutao Meng, Xujia |
| author_facet | Ma, Yutao Meng, Xujia |
| contents | Consider the complex Ginibre ensemble, whose eigenvalues are $(λ_i)_{1\le i\le n}$ and the spectral radius $R_n=\max_{1\le i\le n}|λ_i|.$ Set $X_n=\sqrt{4 γ_{n}}(R_{n}-\sqrt{n}-\frac12\sqrt{γ_{n}})$ and $F_n$ be its distribution function, where $γ_{n}=\log n-2\log(\sqrt{2π}\log n).$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $Λ.$ We prove in further in this paper that $$\lim_{n\to\infty} \frac{\log n}{\log\log n}\, W_1\left(F_n, Λ\right)=2$$
and the Berry-Esseen bound
$$\lim\limits_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x\in \mathbb{R}}|F_{n}(x)-e^{-e^{-x}}|=\frac{2}{e}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution Ma, Yutao Meng, Xujia Probability 60F10, 60B20, 60G55 Consider the complex Ginibre ensemble, whose eigenvalues are $(λ_i)_{1\le i\le n}$ and the spectral radius $R_n=\max_{1\le i\le n}|λ_i|.$ Set $X_n=\sqrt{4 γ_{n}}(R_{n}-\sqrt{n}-\frac12\sqrt{γ_{n}})$ and $F_n$ be its distribution function, where $γ_{n}=\log n-2\log(\sqrt{2π}\log n).$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $Λ.$ We prove in further in this paper that $$\lim_{n\to\infty} \frac{\log n}{\log\log n}\, W_1\left(F_n, Λ\right)=2$$ and the Berry-Esseen bound $$\lim\limits_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x\in \mathbb{R}}|F_{n}(x)-e^{-e^{-x}}|=\frac{2}{e}.$$ |
| title | Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution |
| topic | Probability 60F10, 60B20, 60G55 |
| url | https://arxiv.org/abs/2501.08039 |